Identification of influential users in social network using gray wolf optimization algorithm

Identification of influential users in social network using gray wolf optimization algorithm

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Identification of influential users in Social Network Using Grey Wolf Optimization Algorithm Ahmad Zareie ConceptualizationSoftwareWriting – Review & EditingProofread , Amir Sheikhahmadi ConceptualizationMethodologySoftwareWriting- Original DraftSupervisionProofread , Mahdi Jalili MethodologySoftwareWriting- Original DraftWriting – Review & EditingProofread PII: DOI: Reference:

S0957-4174(19)30689-X https://doi.org/10.1016/j.eswa.2019.112971 ESWA 112971

To appear in:

Expert Systems With Applications

Received date: Revised date: Accepted date:

17 May 2019 7 September 2019 19 September 2019

Please cite this article as: Ahmad Zareie ConceptualizationSoftwareWriting – Review & EditingProofread , Amir Sheikhahmadi ConceptualizationMethodologySoftwareWriting- Original DraftSupervisionProofread , Mahdi Jalili MethodologySoftwareWriting- Original DraftWriting – Review & EditingProofread , Identification of influential users in Social Network Using Grey Wolf Optimization Algorithm, Expert Systems With Applications (2019), doi: https://doi.org/10.1016/j.eswa.2019.112971

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Identification of influential users in Social Network Using Grey Wolf Optimization Algorithm Ahmad Zareie1, Amir Sheikhahmadi1,* , Mahdi Jalili2 1

Department of Computer Engineering, Sanandaj Branch, Islamic Azad University, Sanandaj, Iran

2

School of Engineering, RMIT University, Melbourne, Australia

*

Corresponding author: Amir Sheikhahmadi Tel.: +98 9183735284.

E-mail addresses: [email protected] (Ahmad Zareie), [email protected] (Amir Sheikhahmadi), [email protected] (Mahdi Jalili).

Abstract A challenging issue in viral marketing is to effectively identify a set of influential users. By sending the advertising messages to this set, one can reach out the largest area of the network. In this paper, we formulate the influence maximization problem as an optimization problem with cost functions as the influentiality of the nodes and the distance between them. Maximizing the distance between the seed nodes guarantees reaching to different parts of the network. We use grey wolf optimization algorithm to solve the problem. Our experimental results on three real-world networks show that proposed method outperforms state-of-the-art influence maximization algorithms. Furthermore, it has lower computational time than other meta-heuristic methods. Keywords: Influence maximization, Grey wolf Optimizer, Viral Marketing, Spreading process, Social Networks

1. Introduction Survival of many business firms depends on the presentation and sale of their products. For this purpose, they are always looking for ways to effectively represent their products in the marketplace. Word of mouth and propagating the advertisement messages in an appropriate context by people selected properly, may achieve this goal. Online social networks are *

Corresponding author. Email address: [email protected]

appropriate choices for propagating advertisements of products. Engrossing environments of these networks have attracted many users, which are on increase every day (Kimura, Saito, Nakano, & Motoda, 2010; Sheikhahmadi, Nematbakhsh, & Zareie, 2017; Zareie, Sheikhahmadi, & Jalili, 2019a). Viral marketing can much benefit from properties of online social networks, where a set of users, referred to as seed set, is selected to initiate spreading the messages on the network. Due to the limitation of advertising budget of firms, there is only a limited number of members of the seed (Chevalier & Mayzlin, 2006; Zareie, Sheikhahmadi, & Khamforoosh, 2018). A question arises here that is how to effectively identify the most influential users to place them into the seed set. This problem is known as Influence Maximization (IM) problem, that is defined as to identify a set of k users on which the spreading process is initiated (Lu et al., 2017; Sheikhahmadi, Nematbakhsh, & Shokrollahi, 2015). Although in some approaches, e.g. (Sheikhahmadi et al., 2017; Zareie, Sheikhahmadi, & Jalili, 2019b), users' behavior and interests have been taken into account as important factors in the spreading process, such information are not always available in real networks. Thus, we often have only network structural information available to identify influential users. A number of diffusion models (Huang, Lee, Wen, & Sun, 2013; Jalili & Perc, 2017; Nowzari, Preciado, & Pappas, 2016) have been proposed to simulate the spreading process and measure the influentiality of the seed set. Due to large search space of the problem, assessing all possible subsets to find the optimal seed set is an NP-hard problem (Kempe, Kleinberg, & Tardos, 2003). Real networks are often large-scale, and thus applying diffusion models to measure the influentiality of all users can be a time consuming process. A possible way to identify the most influential users is to choose the most central nodes. Structural centrality can be obtained using various measures, such as degree (Freeman, 1978), betweenness (Freeman, 1977), closeness (Sabidussi, 1966), K-shell (Kitsak et al., 2010), and hierarchical K-shell (Zareie & Sheikhahmadi, 2018). However, structural centrality measures often do not result in near optimal solution to the problem (Bao, Liu, & Zhang, 2017; Zareie et al., 2018). Some other research studies have formulated IM problem as an optimization problem, used greedy, heuristic, and meta-heuristic methods to solve it. Greedy methods (Kempe et al., 2003; Y. Wang, Cong, Song, & Xie, 2010) select a solution close to the optimal solution by using a diffusion model. Although they find highly accurate solutions, they suffer from high computational complexity, making them impractical on large networks (Gong, Yan, Shen, Ma, & Cai, 2016). Heuristic methods (Bao et al., 2017; Zareie et al., 2018) proposed for IM

problem compromise the accuracy by reducing computational complexity. They are also likely to trap in a local optima (Cui et al., 2018). A number of recent works (Cui et al., 2018; Gong et al., 2016) proposed evolutionary optimization methods to avoid this. These research studies have been carried out to identify near-optimal solution of the problem. To find a near-optimal set of the influential users, one needs to first measure the influence of users in an effective manner, and then find the most influential users. In this work we first use a metric based on entropy to measure influentiality of users. We then formulate the problem as an optimization problem and use grey wolf optimization method (Mirjalili, Mirjalili, & Lewis, 2014) to solve it. In comparison to other evolutionary optimization algorithm, this algorithm has less tunable parameters, and can search the problem space with less computational complexity and faster convergence time (Jayakumar, Subramanian, Ganesan, & Elanchezhian, 2016; Pradhan, Roy, & Pal, 2016). The rest of the paper is organized as follow. Section 2 covers the review of diffusion models and previous works. Preliminary information of the proposed method is introduced in section 3. Section 4 is dedicated to the formal definition of the problem and the details of the proposed method. Details of the experiments and the results of the evaluation of the proposed method are reported in section 5, and finally section 6 concludes the paper.

2. Diffusion models and related works Diffusion models are used to simulate the spreading process in real world. Some popular diffusion models are discussed in this section. In addition, a review of well-known methods for IM problem is presented in this section. A network is modeled as a graph where the nodes represent the network users, and the relationships between the users are shown by edges. 2-1. Diffusion models Diffusion models are used to simulate the spreading process in real world. Diffusion models are generally divided into threshold (Granovetter, 1978; Kempe et al., 2003), cascading (Goldenberg, Libai, & Muller, 2001; Kempe et al., 2003), and epidemic (Buscarino, Fortuna, Frasca, & Latora, 2008; Peng, Yu, & Yang, 2014) models. Independent cascading (IC) model (Goldenberg et al., 2001) is one of the most popular models widely used in the literature. In IC model, each user can be in either active or inactive modes. In order to examine the influentiality of a seed set S, the nodes placed in S are initially set as active users and all other users are considered as inactive. At each time step t, each user activated in step t-1 has a

single chance to activate each of its inactive neighbors with activation probability p. This process continues until no new user becomes active in a time step. At the end, the number of activated users during the process is considered as the influence spread of S. In order to obtain statistically significant results, it is repeated many times, and the average number of activated users is considered as influentiality of S. 2-2. Related works The existing research studies on IM problem can be classified into two main categories. The first category identify influentiality of users and rank them. In most of these studies, according to the network structure and users’ position in the network, a centrality measure is used to measure the influentiality of the users. Degree centrality (Freeman, 1978), betweenness (Freeman, 1977), closeness (Sabidussi, 1966), k-shell (Kitsak et al., 2010), Page Rank (Brin & Page, 2012), neighborhood diversity (Zareie et al., 2019b), and hierarchical kshell (Zareie & Sheikhahmadi, 2018) are some well-known centrality measures used for this purpose. Although classical centrality measures that are simple to computing, can determine node vitality (Davoudi & Chatterjee, 2018; Kitsak et al., 2010; Meo, Musial-Gabrys, Rosaci, Sarne, & Aroyo, 2017; Yuan, Sun, & Li, 2017; Zareie & Sheikhahmadi, 2018), they do not often result in high levels of accuracy in correctly identifying top-k most influential nodes. A number of measures have been proposed to optimally select seed nodes maximizing their collective influence range. One may improve further the performance, by considering not only influentiality of nodes, but also distance between them on the network (Bao et al., 2017). In order to obtain maximum collective influentiality for k selected nodes, they should be influential nodes and at the same time located in different parts of the network, i.e. distance between them is maximized (Zareie et al., 2018). This guarantees maximizing reachability of larger part of the network when the seed nodes are first activated. The second category of IM methods consider this and formulate the problem as an optimization problem. The optimization-based methods are classified into three groups, which are discussed in the following. 2-2-1. Greedy methods In greedy method (Kempe et al., 2003), identification of a seed set S containing k most influential users is iterated in k steps and diffusion models are applied to estimate influentiality of the set at each iteration. First, S is considered as an empty set. Then, the most influential user is identified and added to S as the first member. In the second step, the

influentiality of S + {v} is determined for every



and the user whose union with S

has the highest influence, is added to S as the new member. By iterating this process, k nodes are added to S. The selected set by this method has an acceptable influence spread, but given the high time complexity, applying this method is time consuming in large networks. If the diffusion model used for numerical simulation is sub-modular, one can used methods like Cost efficient lazy forward (CELF) (Leskovec et al., 2007) to reduce the time complexity of the greedy method. Although this method is much faster than the greedy one, it still suffers from a high time overload. CELF++ (Goyal, Lu, & Lakshmanan, 2011) is an improvement over the CELF, which improves its time complexity up to 35 to 55. Likewise, various methods (Heidari, Asadpour, & Faili, 2015; Y. Wang et al., 2010) have been proposed to improve the greedy method. 2-2-2. Heuristic methods In order to improve the time complexity of the optimization process of IM problem, a series of heuristic methods have been presented for selecting the near-optimal set of influential users. In these methods, a score is assigned to each user by using a recursive approach, and the users with the highest score are selected as seeds. In (Chen, Wang, & Yang, 2009), two methods, named single discount and double discount, were presented for selecting the seed set. In these methods, the degree of each node is initially considered as indicator of its influentiality, and selection of seeds is iterated in k steps. At each step, the node with the highest influentiality is added to the seed set as a new member. If node v is added to the set as a new member, the influentiality of its neighbors is reduced, and they will subsequently have less chance to be selected as the seed. Degree punishment method (X. Wang, Su, Zhao, & Yi, 2016) also tries to select the seeds by the same approach. In this method, if node v is selected as a seed, the influentiality of its neighbors and second-order neighbors is punished, and their chance to be selected as seed is reduced. This strategy leads to less overlap and more dispersion of the seeds in the network. In degree distance method (Sheikhahmadi et al., 2015), identification of influential users is similarly repeated in k steps, where at each step, the node with the highest degree is considered as a candidate to be added to the seed set as a new member. If there is an acceptable distance between the candidate and each member of the seed set, it is added to the set as a new member; otherwise the node with the next highest degree is nominated as the candidate. In (Guo, Lin, Guo, & Liu, 2016), a distance-based graph coloring method is presented to identify the seeds with proper distance to each other. This method tries to classify the nodes so that the distance between each pair of the nodes in

each group is higher than a threshold. In (Bao et al., 2017), a heuristic clustering method was presented, where the similarity between each pair of nodes is first determined, and nodes are clustered in k different clusters. Then, the most influential node in each cluster is selected as members of the seed. IM problem is modeled as a multi-objective optimization problem and a heuristic model is proposed to solve the problem (Zareie et al., 2018). In spite of low time complexity, the heuristic methods suffer from low accuracy in selecting the near-optimal set of influential nodes (Cui et al., 2018), which is mainly due to being trapped in a local optimal solution. 2-2-3. Meta-heuristic methods In these methods, by defining a fitness function, IM problem is modeled as an optimization problem, and methods like evolutionary optimization algorithms are applied to solve it. In (Jiang et al., 2011), the influentiality of the seed set S is defined as Expected Diffusion Value (EDV), as:

( )



(

(

)

( ))

(1)

( )

where

( )

members of

is the neighbors of the members of S and ( )

( )

denotes the set which are

but not members of S. p shows the activation probability. In spreading ( )

process, each node

will not be activated with probability (

)

( )

, where t(u)

is the number of neighbors of node u which are members of S, it will be accordingly activated with probability

(

)

( )

. Thus, the size of the seed set, k, as well as the number of

neighbors being activated in the spreading process, is considered as in Eq. (1). The simulated annealing algorithm was used to find S with the aim of maximizing EDV. In (Gong et al., 2016), an improvement of EDV is introduced, and Particle Swarm Optimization (PSO) algorithm is used to maximize the objective function. (Cui et al., 2018) proposed a method find S by using differential evolution algorithm with the aim of maximizing the EDV function. Replica symmetrically mean-field theory is used to solve IM problem in (Sun, 2016). Apart from nodes' influence, budget constraint can also be taken into account to define the IM as an optimization problem (Borrero, Prokopyev, & Krokhmal, 2018; Eshghi et al., 2018; Preciado, Zargham, Enyioha, Jadbabaie, & Pappas, 2013).

The overlap between the members of the seed has been neglected in all of the above mentioned meta-heuristic methods, and the selected seeds may cover limited numbers of nodes with high activation probabilities. Therefore, in this paper, a fitness function is first defined, and then by using the grey wolf optimization algorithm a method is presented to maximize the fitness function. 3. Preliminary information In the proposed algorithm, by applying the concept of entropy a fitness function is first defined and IM problem is modeled as an optimization problem. The problem is then solved using grey wolf optimization algorithm. The concept of entropy was firstly introduced by Shannon in 1948 for determining the amount of information in an event (Jaynes, 1957). Based on the Shannon entropy, if X is a set of possible events probability of occurrence of the event

such that ∑

, and

is the

, then the entropy of X is

computed using Eq. (2). ( )



( )

(2)

Grey wolf optimization (GWO) Algorithm (Mirjalili et al., 2014) is a population-based evolutionary algorithm inspired the hunting behavior of grey wolves. As shown in Fig. 1, these wolves follow a strict social dominant hierarchy, and are divided into four categories in the society: Alpha (α), Beta (β), Delta (δ), and Omega (ω) wolves.

Figure 1. Social dominant hierarchy of grey wolves group (Mirjalili et al., 2014)

Alpha, Beta, and Delta wolves are responsible for managing the attack in the hunt process, and Omega wolves do not have a specific performance for the group, and they mostly play the role of scapegoat. According to the lifestyle and hunting process of these animals, GWO algorithm is modeled as follows.

First, a set of solutions is randomly generated. Each solution is assigned to a wolf indicating its position. The wolf which has the best fitness is considered as the Alpha, and the wolves having the second and third best fitness are considered as Beta and Delta wolves, respectively. The other wolves are considered as Omega wolves. In this algorithm, the problem space is searched according to the position of Alpha, Beta, and Delta wolves to find the position of the prey (optimal solution). In other words, these three wolves estimate the position of the prey, and the omega wolves update their position based on the position of these wolves to find a closer position to the prey. The algorithm attempts to find the optimal solution during iterations. In each iteration, Omega wolves try to improve their position according to the position of the other dominant wolves. In each iteration t, Omega wolf i (

determines its new position for iteration t+1, ⃗(

)

⃗⃗

), as Eq. (3). ⃗⃗

⃗⃗

(3)

That is to say each omega wolf i seeks to find a better position based on the three values ⃗⃗

⃗⃗ and ⃗⃗ , where the value ⃗⃗ is calculated according to the current position of the wolf

and the position of alpha wolf, as Eq. (4). ⃗⃗



⃗ ⃗⃗

⃗⃗

|⃗ ⃗

⃗ ( )|

(4)

where t represents the current iteration, and ⃗ ( ) expresses the position of Omega wolf i in this iteration. ⃗⃗⃗ indicates the positions of Alpha. Using Eqs. (5) and (6) the values of ⃗⃗ and ⃗⃗ are calculated according to the current position of the wolf and that of beta wolf and delta wolf, respectively. ⃗⃗ ⃗⃗



⃗ ⃗⃗ ⃗

⃗⃗

⃗ ⃗⃗

⃗⃗

|⃗ ⃗ |⃗ ⃗

⃗ ( )| ⃗ ( )|

(5) (6)

where ⃗ and ⃗ are the position of beta and delta wolves, respectively. A and C are coefficient vectors, which are calculated by Eqs. (7) and (8). ⃗

⃗ ⃗⃗

(7) ⃗⃗

(8)

In Eqs. (7) and (8), ⃗ and ⃗

are random values in the range [0, 1], and

is a control

parameter that approaches linearly from 2 to 0 during the iterations. Eq. (9) is used to calculate the value of

in iteration t. (

(9)

)

where max_t shows the number of iterations over the algorithm. Over the course of the iterations, the contradiction among the position of the wolves reduces, and the algorithm converges. At the end of the algorithm, the best solution, which is the Alpha wolf, is returned as the optimal solution. The pseudo-code of GWO algorithm is shown in Algorithm 1, in which n represents the size of the population, i.e. the number of wolves. Algorithm 1: Pseudo-code of GWO Algorithm 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

) Initialize the random position of each grey wolf as ( Initialize a, A, and C Calculate the fitness value for each wolf Set , , and as best, the second best, and the third best wolves, respectively Consider the other wolves as Omega wolf Set t = 0 While (t < max_t) For each Omega wolf Update the position of using Eq. (3) End for Update a, A, and C using Eqs. (6) to (8) Calculate the fitness value for each Omega wolf Update , and t=t+1 End while Return as selected solution

4- The proposed IM algorithm A social network can be represented as an undirected graph G = (V, E) where {

} shows the network users, and E shows the relationship between them. Edge represents the edge between two nodes

neighbors. degree of

and

is used to represent the neighbors of node .

( )

; these two nodes are called , and

represents the

shows the second-order neighbors, i.e. neighbor of neighbor, of

. The

goal of IM problem is to identify a set S with k members to initiate the spreading process so that the spread is maximized, i.e. the number of activated users is maximized. In the rest of

the paper,

shows the set of nodes located in the neighborhood or second-order

neighborhood of seed set S. 4-1. fitness function The proposed algorithm select nodes as members of the seed set, such that (i) each node has high influentiality, and (ii) the selected set covers broad parts of the network as much as possible. This guarantees maximum propagation. During the propagation process, the probability that node

( )

where

is activated, i.e. receives the message, is calculated as:



(



shows the probability of propagation from

calculates the probability that node members of S; it is considered 0 if node

to

)

(10)

. The first section of Eq. (10)

receives the message of its neighbors, which are has no neighbors in S. The second section of the

equation calculates the probability that node

receives the message from its second-order

neighbors that are also members of S. These two parts are summed due to the probability rules. Different nodes do not have the same impact during the propagation process. If the nodes with high degree receive the message, the message may be spread to further parts of the network. Therefore, the worthiness of each node ( )

is calculated by:

( )

(11)

Therefore, in the proposed algorithm, called Grey Wolf based Influence Maximization (GWIM), Eq. (12) can be considered as the fitness function. ( )



( )

(12)

Applying summation operator in the fitness function might not be efficient as it cannot ensure the minimum overlap between the seeds. In GWIM, we seek to find the seeds, such that (i) the number of nodes in

is maximized, and (ii) all of these nodes have high worthiness. In

the case of using summation operator, a composition of a limited number of worthy nodes and significant number of unworthy ones in

may maximize the value of the fitness

function (12). In this case, failure in propagation of the message towards the worthy nodes

can lead to dramatic decrease in the influentiality of the selected seeds. Entropy is used in the fitness function to tackle this issue, which indeed determines how uniform distribution of can determine influence spread of S more accurately. On the other

worthiness of nodes in

hand, the value of entropy rises as the size of

increases, and thus a proper normalization

needs to be considered. The entropy of the worthiness as the fitness value of S, is defined as: ( )

( )



( )

(

( )

( ) is the sum of the worthiness of the nodes in

where

( ) ( )

)

(13)

and is calculated using Eq. (12).

4-2. Proposed algorithm According to (Sheikhahmadi et al., 2017), the nodes with degree 1 have a very low chance to be selected as seeds. In many real social networks, there are large portion of nodes with only one neighbors. Therefore, in order to reduce the time complexity of GWIM algorithm, only nodes with degree higher than 1 are considered as possible seeds; let’s denote such nodes as {

}. During the proposed algorithm, each wolf (solution) has two

property: position and corresponding seed set. The position of wolf i is shown as a vector with | | elements, where the j-th indicates the chance of node The corresponding seed set of wolf i is shown as values in

.

to be selected as a seed.

, which contains k nodes with the highest

Algorithm 2: pseudo-code of GWIM Input: Undirected Graph G=(V,E), the seed set size k, the population size n, the number of 01 iterations max_t Output: S // a set with k members as initial seed set 02 03 Begin Algorithm * + 04 Initial a, A, and C and 05 Initial n position to randomly and determine corresponded to each 06 Calculate the fitness for each (i=1 , 2, ..., n) 07 According to the fitness values, select the best solution as , the second best solution as , the third best solution as , and the others as Omega solutions Set t = 0 08 09 While (t
Algorithm 2 shows the pseudo-code of GWIM algorithm. In line 05 of the algorithm, n primary solutions are generated randomly. Random position function is used to generate each random solution. In line 06, the fitness value of the solutions is computed using (12). In line 07, according to the fitness value of the solutions, Alpha, Beta, and Delta wolves are determined. In lines 9-17, the operations are iterated in max_t times to maximize the fitness function of the wolves. In each iteration, by using Update position function, the position of Omega wolves is updated, and given to the updated positions. Seed set each wolf i in lines 10-12. In line 14, the fitness value is calculated for each

is determined for , and according

to the fitness value, the best solutions are considered as new Alpha, Beta, and Delta wolves. In line 15, the position of Beta and Delta wolves are randomly regenerated, if or

is equal to

is equal to

. This helps avoiding being trapped in local optimal solution. The

functions used in Algorithm 1 are further described below. 4-2-1. Generation of primary population In GWIM algorithm, Random position function is used to generate a primary solution i (primary position

of wolf i and its corresponding seed set ). The pseudo-code of this

function is shown in Algorithm 3. In this algorithm, a random value

is given to each node

, as its chance of being selected as a seed, which is proportional to their degree (line 04). k nodes corresponding to the k highest entries of Xi are chosen as

members in lines 08

to 11. Algorithm 3: Random position function 01 02 03 04 05 06 07 08 09 10 11 12 13

Input: Undirected Graph G = (V,E), , the seed set size k. Output: , the position of wolf i, and , corresponded seed set. For j = 1 to r = rand (1... ) max(d) End for Set *+ For i = 1 to k Find the next maximum Add to End for Return and End Algorithm

4-2-2: Updating Wolves’ position In GWIM algorithm, according to the position of Alpha, Beta and Delta wolves, the position of each Omega wolf is updated with the hope of achieving a better position. Update position function is defined for this purpose. The pseudo-code of this function is shown in Algorithm 4. In this algorithm, in order to update the chance of node

in wolf i,

node in Alpha wolf,

, is used. Therefore, the values

of

,

,and

, Beta wolf,

are calculated for node

calculated. By calculating the value of i for iteration t+1,

, and Delta wolf,

(

(

, and the value of ) for each node

(

, the chance of this

) is accordingly the position of wolf

), is updated. In algorithm 4, Abs(a) shows the absolute value of a.

Algorithm 4: Update position function 01 02 03 04

Input: A, C , , ( ), Output: ( ) For j=1 to

,

and //updated

(

( ))

(

( ))

(

( ))

(Xij is the j-th entry of Xi)

05 06 07 08 09 10

(

)

11 12 13

End for Return ( End Algorithm

)

For example, consider a network in which nodes 1, 4, 5, 7, 15, 17, 21, 29, 30, 40 have degree *

greater than 1, and we fix k = 3. Here,

+ and

thus the length of each position is 10. Let’s suppose the position vector

10, and

assigned to wolf i

as: 1

4

5

7

15

17

21

29

30

40

0.336 0.215 0.101 0.156 0.546 0.489 0.112 0.356 0.686 0.125

Each

in the position shows the chance of jth node in

to be selected as the seed. For

example, the chance of node 1 is 0.336, while it is 0.215 for node 4. k nodes with the highest chances are selected as the corresponding solution that is

*

+. Consider that

max_t = 10 and we want to update the position of wolf i in the sixth iteration, i.e. t = 6. a is calculated as

(

)

. ⃗ and ⃗ are two matrices with 3 rows and

columns. A random value in the range [0, 1] is generated for ⃗ . Suppose ⃗ the first entry of vector ⃗ (that is ⃗ ), is calculated as

. Thus, . This

process is repeated to update all entries of ⃗. Matrix ⃗ is updated using a similar procedure

based on random values in ⃗ (see Eq. (8)). Let’s suppose that the position and corresponding solutions for alpha, beta and delta wolves are as follow: Wolf

Alpha

Position 1



4

5

7

15

Solution

17

21

29

30

40

*

+

*

+

*

+

0.412 0.185 0.102 0.125 0.564 0.465 0.102 0.359 0.325 0.119

Beta

1



4

5

7

15

17

21

29

30

40

0.512 0.215 0.115 0.112 0.602 0.325 0.215 0.289 0.402 0.190

Delta

1



4

5

7

15

17

21

29

30

40

0.325 0.300 0.365 0.075 0.300 0.352 0.236 0.490 0.458 0.250

The value of ⃗⃗

⃗⃗

and ⃗⃗ are then calculated to update the first entry of

Let’s suppose that ⃗









then have: ⃗⃗

|⃗ ⃗⃗

⃗⃗

⃗ ⃗

|⃗ ⃗⃗

⃗⃗



⃗ ⃗

⃗⃗

⃗ ⃗

⃗⃗



⃗⃗

)

(

)

⃗ ( )| ⃗⃗

Thus, the updated first entry of ⃗ ( )

(

⃗ ( )| ⃗

|⃗ ⃗⃗

⃗ ( )|

⃗⃗

(

)

in the 6th iteration is calculated as follow: ⃗⃗

(that is ⃗

). . We

This process is repeated for each ⃗ ( ), and the position of wolf i is updated in the 6th iteration. This calculations are repeated to update the position of all wolves in each iteration. 4-2. Computational complexity analysis The proposed algorithm has two main steps: the initialization step (lines 4 to 8) and the main (

step (lines 9 to 17). Complexity of line 4 is (

. Having the complexity

) due to assessing of all nodes to determine

), the random position function is called for n

times in line 5. Thus, the complexity of line 5 is

(

). The fitness value is ( 〈 〉〈

calculated for n solutions in line 6 using Eq. (12), with the complexity where 〈 〉 and 〈

( )

( )〉

),

〉 are the average degree and second-order degree of nodes. Thus,

complexity of line 6 is

〈 〉〈

(

( )〉

). The identification of three best solutions can be

done with complexity ( ) in line 7. Thus, the computational complexity of the initialization step is

〈 〉〈

( 〈 〉〈

(

( )〉

( )



), which can be reduced to

).

In lines 10-12, the position of each omega wolf is updated using the Update position function, which has the complexity

(

) Thus, the complexity of lines 10-12 is

(

). As it

was mentioned, the complexity of the calculation of n solutions in line 14 and the identification of three best solutions in line 15 are

〈 〉〈

(

( )〉

) and

The complexity of the main step of the proposed algorithm is 〈 〉〈

( )〉

)), which can be reduced to

overall time complexity of the proposed algorithm is 〈 〉〈

( )〉

(

〈 〉〈 〈 〉〈

( )〉

( )〉

( 〈 〉〈

(

(

( ), respectively. ( ( ) 〉)

). The

(

), which can be reduced to

).

5. Experiments and evaluation of the proposed method For the experiments, three real-world networks are used and their details are shown in Table 1. In this table, p is the activation probability used in IC model. It is set according to the sparsity of the graphs and is calculated based on the average of the degree and second-order degrees of nodes in the network.

Table 1. Details of the real-world networks used in the experiments; |V| is the number of nodes, |E| is the number of edges, 〈d〉 is the average degree of nodes, and p is activation probability used in independent cascade diffusion model.

Network

|V|

|E|

〈 d〉

p

Hamsterster (HAM)full

2,426

16,631

13.711

0.03

Pretty Good (PGP)Privacy

10,680

24,316

4.5536

0.06

Astro (AST)

18,771

198,050

21.102

0.02

In order to assess performance of the proposed method, the results are compared with a number of state-of-the-art influence maximization methods, including: o Centrality measure: ● Degree centrality (Freeman, 1978) ● k-shell (Kitsak et al., 2010) (Kitsak et al., 2010) ● Page Rank (Brin & Page, 2012) ● Betweenness centrality (BC) (Freeman, 1977) o Greedy method: ● CELF++ (Goyal et al., 2011) o Heuristic methods: ● double discount (DD) (Chen et al., 2009) ● heuristic clustering (HC) (Bao et al., 2017) o Meta-heuristic methods: ● Degree Descending Search Strategy (DDSE) (Cui et al., 2018) ● Simulated annealing EDV (SADV) (Jiang et al., 2011)

The experiments are repeated 20 times, and the results show averages over these runs. 5-1. Parameters of GWO In the first experiment, the effects of parameters of GWO on the final value of the fitness function is examined, and the optimal value for the population size n and the number of iteration max_t is achieved. To analyze the impact of max_t, population size n is set 50, and the value of max_t is varied from 1 to 200. The results of this experiment on various networks for k = 20, 40, 60, 80 are shown in Fig. 2. As can be seen, as max_t exceeds 100, the value of the fitness function does not increase significantly; hence, in the next experiments, the value of max_t is set to 100.

Figure 2. The fitness function as a function of max_t in different networks.

In the next experiment, the effect of population size n on the value of fitness is investigated. To do so, the maximum iteration max_t is set at 100, and the value of the obtained fitness in different iterations for population sizes of 10, 30, 50, 70, 90 is reported. The results show that the value of the fitness function increases by increasing the population size. However, no significant increase is obtained when the population size takes values higher than 50, and thus the population size is fixed at 50 for the following experiments.

Figure 3. The fitness function as a function of iterations for different population size n.

5-2. Convergence speed The convergence speed of the proposed method is investigated in this section. To this end so, the movement of the wolves during the optimization process is studied. For this, the Euclidian distance between the positions of each wolf i in two sequential iterations t and t+1 is calculated using:

(

)

√∑ ( ⃗ (

)

⃗ ( ))

(14)

where ⃗ ( )shows the value of jth entry of the position of wolf i in iteration t. The average movement of all wolves in two sequential iterations t and t+1 is calculated as:

(

)



(

)

(15)

where n is the size of population In this experiment, the size of seed set is considered as k = 40. Fig. 5 illustrates the results. As can be seen, the average movement increases during the initial iterations. As the number of iterations increases, the average movement of the wolf decreases, and the algorithm converges. It is also seen that in spite of oscillatory behavior in the final iterations, which is caused by line 15 in algorithm 2 to avoid being trapped in local optimal solution, the proposed algorithm experiences a convergence trend in the final iterations.

Figure 4. The average movement of wolves as a function of the iterations.

5-3. Evaluation of influentiality of the seeds obtained by the proposed method We next evaluate the performance of GWIM with other algorithms by comparing the influentiality of the seeds selected by these algorithms. For this, the size of seed set is varied from 10 to 80. IC model is applied to evaluate the influentiality of each seed set offered by different algorithms. In order to improve the confidence of the numerical results, IC model is repeated 50 times in the experiments of this section. The obtained results shown in Fig. 5 express that GWIM and CELF++ methods explore the search space better than other methods, and results in a highest influentiality for the same seed size. GWIM provides close

results to CELF++, and especially, for greater size of seed set, GWIM explores the search space better than CELF++, and outperforms it. Considering the distance between seeds plays more important role as seed set size increases, and so selected seed set by GWIM propose higher influentiality than other ones. Expecting, as the seed size increases, the influentiality of IM algorithms also increases. DDSE that is another method that uses an optimization process is the third top-performer after GWIM and CELF++.

Figure 5. The influentiality (the number of activated nodes divided by the network size) as a function of size of the seed set. The algorithms including the proposed algorithm (GWIM) are applied on three networks.

In the next experiment, we examine the effect of varying the activation probability of IC model on the performance of the algorithms. To do so, the value of k is set to 40, and the value of the activation probability is varied from 0.01 to 0.1 with interval 0.01. The results of this experiment are shown in Fig. 6, where by increasing p, the influentiality also increases. Almost for all cases, the proposed algorithm is the top-performer, and its outperformance to other algorithms is more pronounced as p increases. The worthiness of nodes increases as p increases, and applying entropy notation in the fitness function leads to select seeds with less

overlap in GWIM. As a result, GWIM outperforms the other algorithms in higher activation probability.

Figure 6. The influentiality (the number of activated nodes divided by the network size) as a function of activation probability. The algorithms including the proposed algorithm (GWIM) are applied on three networks.

5-3. Statistical tests In this section, the differences in performance of the algorithms are tested for statistical significance. To make statistical significance, we conduct Friedman test analysis and use Bonferroni method to correct for multiple comparisons. The confidence score is set as α = 0.05, which means that the differences are statically significate if the p-value is less than 0.05. The p-values are reported in Table 2. In this table, the method with the best results in each value of k is shown by '_' and is accordingly considered as control method. Boldface values indicate that the corresponding hypothesis are accepted. For each value of k, the algorithms are ascendingly ordered based on their p-values. If the ith p-value in the ordered

list is lower than α/ (N-i), where N is the number of the algorithms, then the corresponding hypothesis is rejected; otherwise it is accepted. As can be seen from the table, for the values of k as 10, 20 and 30, CELF++ has the best results and is considered as control method. For k = 10, some algorithms including SWIM provide close performance to CELF++. For the other values of k, GWIM obtains the best results and is considered as the control method. Furthermore, DDSE and CELF++ obtain close performance to GWIM, while CELF++ outperforms DDSE. Method Degree BC k-shell DD Page Rank HC SADV CELF++ DDSE GWIM

k =10 0.007 0.000 0.000 1.000 0.023 0.007 0.255 _ 1.000 1.000

k =20 0.000 0.000 0.000 0.158 0.000 0.001 0.005 _ 1.000 1.000

k =30 0.000 0.000 0.000 0.000 0.000 0.000 0.001 _ 0.643 1.000

k =40 0.000 0.000 0.000 0.001 0.002 0.004 0.010 1.000 1.000 _

k =50 0.000 0.000 0.000 0.000 0.000 0.004 0.030 1.000 1.000 _

k =60 0.000 0.000 0.000 0.000 0.000 0.000 0.065 1.000 1.000 _

k=70 0.000 0.000 0.000 0.000 0.000 0.001 0.032 1.000 1.000 _

k=80 0.000 0.000 0.000 0.000 0.000 0.001 0.000 1.000 0.503 _

5-4. Time complexity We next evaluate efficiency of the algorithms. To this end, the average execution time of the algorithms is compared (Fig. 7). GWIM is more complex than centrality-based methods, but it is more computationally efficient than other meta-heuristic methods, while also resulting in better performance in terms of the influentiality of the obtained seed set. CELF++, has the worst time complexity compared to other methods.

Figure 7. The average run time of the algorithms.

6. Conclusion Effective propagation of information in network systems has important implications in many applications, such as e-commerce and media. Under limited budget, a major challenge in viral marketing is to identify small number of influential users, called seed set. By initially passing the information to the seed set, one might expect to obtain large effect in the network, as the nodes of the seed set can influence others through their connections. This problem is referred to as influence maximization problem is the literature. The influence maximization problem can be modelled as an optimization problem with cost functions such as the influentiality of the nodes and the distance between them. In this paper, we used grey wolf optimization algorithm, as a population-based optimization method, for the influence maximization problem. Our experiments on three real-world networks showed that the proposed algorithm outperforms a number of state-of-the-art influence maximization algorithms. Also, it is not only more effective than other meta-heuristic methods, but also has less computational time. The proposed method has some limitations, which is mainly due to applying populationbased optimization approach. Although it tries to escape from local minima, it can still trap in a local minima, meaning that the final solution will be a local minima rather than a global minima. However, our simulations show that it still results in better performance than stateof-the art.

Declaration of Interest Statement The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

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